Comments on: Complex Roots of Unity

By: Peter L. Griffiths

Peter L. Griffiths — Sun, 11 Mar 2012 18:04:22 +0000

My comments of 23 August 2011 have important implications for Hamilton’s Quaternions equation which is i^2=j^2=k^2=ijk=-1. This equation is incorrect because -1 can only have two square roots, so that k^2 cannot equal -1 unless k is either i or j. Any real, imaginary or complex number can only have two square roots, three cube roots, four fourth roots, five fifth roots etc.

By: admin

admin — Thu, 25 Aug 2011 09:46:30 +0000

@Peter L. Griffiths
Hi, Thanks for taking the time to comment
The method you outlined is one of the easiest I’ve seem for finding the roots of a complex number

By: Peter L. Griffiths

Peter L. Griffiths — Tue, 23 Aug 2011 15:57:29 +0000

It is in fact easier to find the n nth roots of any complex number a+ib. You obtain the arcotangent below 90 degrees of the complex ratio a/b and then divide by the root required. The cotangent of the result gives one of the n roots of the complex ratio. For the other n-1 roots you add 360 degrees to the arcotangent n-1 times. This gives you the n nth roots of the complex number. This system can be reconciled with finding the n nth roots of +1, -1, i and -i by making a in the complex number equal 0 and substituting Cotes’s format cos90+isin90 equals 0+i. It will be found that cos180+isin180 equals -1 and cos360+isin 360 equals i to the power of 4 which is +1. This system also applies for division, cos45+isin45 equals the square root of i.